find the surface area of a regular polygon inscribed in a triangular prism in typescript

To find the surface area of a regular polygon inscribed in a triangular prism, we need to first find the area of the triangular prism and the area of the regular polygon.

Area of the Triangular Prism

The formula for the surface area of a triangular prism is given by:

index.ts
SA = 2 * A_base + P * h
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where A_base is the area of the base, P is the perimeter of the base, and h is the height of the prism.

In our case, the triangular prism has a regular polygon inscribed in its base. So, the base of the prism is a regular polygon, and its area can be calculated using the formula:

index.ts
A_base = 0.25 * n * s^2 * cot(pi/n)
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where n is the number of sides of the polygon, s is the length of each side, and cot is the cotangent function.

To find the perimeter of the base, we can simply multiply the number of sides by the length of each side:

index.ts
P = n * s
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Finally, we need to find the height of the triangular prism. Since the regular polygon is inscribed in the base, its apothem (the distance from the center to the midpoint of any side) will be the height of the triangular prism. We can use the formula:

index.ts
a = s / (2 * tan(pi/n))
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to find the apothem.

Once we have calculated all these values, we can use the formula above to find the surface area of the triangular prism.

Area of the Regular Polygon

To find the area of the regular polygon, we can use the formula:

index.ts
A_poly = 0.5 * apothem * perimeter
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where apothem is the distance from the center to the midpoint of any side of the polygon, and perimeter is the total length of all the sides.

Again, we can use the formula:

index.ts
a = s / (2 * tan(pi/n))
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to find the apothem, and the formula:

index.ts
P = n * s
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to find the perimeter.

Once we have calculated all these values, we can use the formula above to find the area of the regular polygon.

Here's the TypeScript code to implement the above calculations:

index.ts
function surfaceArea(n: number, s: number, h: number): number {
  const cot = (angle: number) => 1 / Math.tan(angle);
  const apothem = s / (2 * Math.tan(Math.PI / n));
  const perimeter = n * s;
  const baseArea = 0.25 * n * s ** 2 * cot(Math.PI / n);
  const prismArea = 2 * baseArea + perimeter * h;
  const polygonArea = 0.5 * apothem * perimeter;
  return prismArea + polygonArea;
}
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We can now call this function with the number of sides of the regular polygon, the length of each side, and the height of the triangular prism to get the surface area of the polygon inscribed in the prism.